Ground state solutions for a class of quasilinear Choquard equation with critical growth

Abstract

In this paper, we consider the following quasilinear Choquard equation with critical nonlinearity \[ \begin{cases} -\triangle u+V(x)u-u\triangle u^{2}=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+u^{2(2^{\ast})-2}u,&x\in\mathbb{R}^{N}, \\ u>0,&x\in\mathbb{R}^{N}, \end{cases} \] where \(I_{\alpha}\) is a Riesz potential, \(0<\alpha<N\), and \(\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}\), with \(2^{\ast}=\frac{2N}{N-2}\). Under suitable assumption on \(V\), we research the existence of positive ground state solutions of above equations. Moreover, we consider the ground state solution of the equation (1.4). Our work supplements many existing partial results in the literature.